矩阵的最优秩1 Hankel逼近：Frobenius范数、谱范数和Cadzow算法
Optimal Rank-1 Hankel Approximation of Matrices: Frobenius Norm, Spectral Norm, and Cadzow's Algorithm
摘要：
本文用一种新的方法刻画了具有Hankel或Toeplitz结构的最优秩1矩阵逼近关于两种不同范数的性质，即Frobenius范数和谱范数.更确切地说，我们表明，这些秩1矩阵逼近问题可以解决最大化特殊的有理函数。我们的方法使我们能够表明，这两个规范的最优解具有完全不同的结构，只有在平凡的情况下，当奇异值分解已经提供了一个最佳的秩1近似所需的汉克尔或Toeplitz结构相吻合。我们还证明了Cadzow算法的结构低秩近似总是收敛到一个不动点在秩1的情况下。然而，它通常不收敛到最优解，无论是关于Frobenius范数还是谱范数。
英文摘要：
We characterize optimal rank-1 matrix approximations with Hankel or Toeplitz structure with regard to two different norms, the Frobenius norm and the spectral norm, in a new way. More precisely, we show that these rank-1 matrix approximation problems can be solved by maximizing special rational functions. Our approach enables us to show that the optimal solutions with respect to these two norms have completely different structure and only coincide in the trivial case when the singular value decomposition already provides an optimal rank-1 approximation with the desired Hankel or Toeplitz structure. We also prove that the Cadzow algorithm for structured low-rank approximations always converges to a fixed point in the rank-1 case. However, it usually does not converge to the optimal solution, neither with regard to the Frobenius norm nor the spectral norm.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
